I might point out a couple of things about LED strips: they're very omni-directional. That's great for area lighting, but if you're doing fine work, consider an MR16 3-5W LED downlight globe like from this
supplier (3W MR16 = $2.05). Warm white may be better for close work than the white (almost bluish) that you normally get off "white" LEDs. I have a few of these globes and find them more than adequate. The extra bonus is that they don't give off any heat so you can have it close to your work and your face without drama.
Being such a low wattage also means you don't have to worry so much about cable size. If you have a dual battery, I'd invest in 15m of fig8 8Ga cable like
this from Jaycar and a few 50A Anderson plugs (
here is a seller with 10 Anderson plugs for $27). Run the cable from the dual battery (via a fuse, please, make it about 40A - inline is fine) to the rear of the vehicle and cut the cable so it will go into an Anderson plug that you can screw into the back underneath. Leave a little extra length (5-10cm) just in case and remember to cable-tie it up.
Then make another cable up that goes from the front of the trailer (through the hand loop for the towball) to the car-mounted Anderson plug. This cable runs into the trailer and provides your juice (use a distribution block inside).
The biggest advantage in this is that you can then make a hookup cable to a jumper battery, or to a generator, or a cable with alligator clips to go onto another battery or even a solar setup.